Abstract Computer models of dynamic systems produce outputs that are functions of time; models that solve systems of differential equations often have this character. Time-indexed inputs, such as the functions that describe time-varying boundary conditions, are also common with such models. Morris (2012) described a generalization of the Gaussian process often used to produce “meta-models” when inputs are finite-dimensional vectors, that can be used in the functional input setting, and showed how the maximin distance design optimality criterion (Johnson et al., 1990) can also be extended to this case. This paper describes an upper bound on the maximin distance criterion for functional inputs. A class of designs that are optimal under certain conditions is also presented; while these designs are of limited practical value, they show that the derived bound cannot be improved in the general case.